Abstract

It is shown that, for an arbitrary discrete process embedded in independent additive discrete noise, the classical binary detection problem using a likelihood-ratio test reduces to a simple comparison of the number of events with a single threshold. Only a weak condition on the noise distribution is required. Our results are appropriate for the analysis of photocounting optical communications and photocounting radar systems as well as neural counting in auditory psychophysics. We specifically apply our method to a signal-detection theory model of the human visual system and draw a comparison to the analysis of Hecht, Shlaer, and Pirenne [J. Gen. Physiol. 25, 819 (1942)].

© 1977 Optical Society of America

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References

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  1. S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
    [CrossRef] [PubMed]
  2. M. C. Teich, P. R. Prucnal, “Negative binomial vs Poisson photon counting in visual psychophysics” (to be published).
  3. M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” J. Royal Stat. Soc. A 83255–279 (1920).
    [CrossRef]
  4. L. Mandel, “Fluctuations of photon beams: the distribution of photo-electrons,” Proc. Phys. Soc. 74, 233–242 (1959).
    [CrossRef]
  5. S. Hecht, “Energy and vision,” in Science in Progress, G. A. Baitsell, ed. (Yale U. P., New Haven, 1945), fourth series, pp. 75–95, 309–310.
  6. A. Rose, “The sensitivity performance of the human eye on an absolute scale,” J. Opt. Soc. Am. 38, 196–208 (1948).
    [CrossRef] [PubMed]
  7. W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
    [CrossRef]
  8. W. P. Tanner, J. A. Swets, “The human use of information—I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory IT-4, 213–221 (1954).
    [CrossRef]
  9. W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954).
    [CrossRef] [PubMed]
  10. H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
    [CrossRef] [PubMed]
  11. H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. 136, 469–488 (1957).
    [PubMed]
  12. W. P. Tanner, “Physiological implications of psychophysical data,” Ann. NY Acad. Sci. 89, 752–765 (1961).
    [CrossRef]
  13. W. J. McGill, “Neural counting mechanisms and energy detection in audition,” J. Math. Psychol. 4, 351–376 (1967).
    [CrossRef]
  14. B. Sakitt, “Counting every quantum,” J. Physiol. 223, 131–150 (1972).
    [PubMed]
  15. B. Sakitt, “Canonical ratings,” Percept. Psychophys. 16, 478–488 (1974).
    [CrossRef]
  16. B. Reiffen, H. Sherman, “An optimum demodulator for Poisson processes: photon source detectors,” Proc. IEEE 51, 1316–1320 (1963).
    [CrossRef]
  17. M. C. Teich, R. Y. Yen, “Photocounting receiver performance for detection of multimode laser or scattered radiation,” IEEE Trans. Aerosp. Electron. Syst. AES-8, 13–18 (1972).
    [CrossRef]
  18. G. W. Flint, “Analysis and optimization of laser ranging techniques,” IEEE Trans. Mil. Electron. MIL-8, 22–28 (1964).
    [CrossRef]
  19. J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  20. M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).Phys. Rev. Lett. 36, 1473 (1976).
    [CrossRef]
  21. H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968), Part I.
  22. J. P. Egan, Signal Detection Theory and ROC Analysis (Academic, New York, 1975).
  23. This calculation may also be carried out by using a special case of Eq. (7. 6. 2) along with the identity (C. 3. 1), in R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).
  24. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, New York, 1965), 4th ed., p. xxxii.
  25. C. Freed, H. A. Haus, “Photoelectron statistics produced by a laser operating below the threshold of oscillation,” Phys. Rev. Lett. 15, 943–946 (1965);“Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. QE-2, 190–195 (1966).
    [CrossRef]
  26. A. W. Smith, J. A. Armstrong, “Observation of photon counting distribution for laser light below threshold,” Phys. Lett. 19, 650–651 (1966).
    [CrossRef]
  27. W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);“Intensity fluctuations in light beams with several degrees of freedom,” Phys. Rev. Lett. 16, 531–533 (1966).
    [CrossRef]
  28. F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
    [CrossRef]
  29. M. C. Teich, P. R. Prucnal, “Photon counting and energy detection in vision” (to be published).

1976 (1)

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).Phys. Rev. Lett. 36, 1473 (1976).
[CrossRef]

1974 (1)

B. Sakitt, “Canonical ratings,” Percept. Psychophys. 16, 478–488 (1974).
[CrossRef]

1972 (2)

B. Sakitt, “Counting every quantum,” J. Physiol. 223, 131–150 (1972).
[PubMed]

M. C. Teich, R. Y. Yen, “Photocounting receiver performance for detection of multimode laser or scattered radiation,” IEEE Trans. Aerosp. Electron. Syst. AES-8, 13–18 (1972).
[CrossRef]

1967 (1)

W. J. McGill, “Neural counting mechanisms and energy detection in audition,” J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

1966 (1)

A. W. Smith, J. A. Armstrong, “Observation of photon counting distribution for laser light below threshold,” Phys. Lett. 19, 650–651 (1966).
[CrossRef]

1965 (3)

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[CrossRef]

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

C. Freed, H. A. Haus, “Photoelectron statistics produced by a laser operating below the threshold of oscillation,” Phys. Rev. Lett. 15, 943–946 (1965);“Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. QE-2, 190–195 (1966).
[CrossRef]

1964 (2)

G. W. Flint, “Analysis and optimization of laser ranging techniques,” IEEE Trans. Mil. Electron. MIL-8, 22–28 (1964).
[CrossRef]

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);“Intensity fluctuations in light beams with several degrees of freedom,” Phys. Rev. Lett. 16, 531–533 (1966).
[CrossRef]

1963 (1)

B. Reiffen, H. Sherman, “An optimum demodulator for Poisson processes: photon source detectors,” Proc. IEEE 51, 1316–1320 (1963).
[CrossRef]

1961 (1)

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. NY Acad. Sci. 89, 752–765 (1961).
[CrossRef]

1959 (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photo-electrons,” Proc. Phys. Soc. 74, 233–242 (1959).
[CrossRef]

1957 (1)

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. 136, 469–488 (1957).
[PubMed]

1956 (1)

1954 (3)

W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
[CrossRef]

W. P. Tanner, J. A. Swets, “The human use of information—I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory IT-4, 213–221 (1954).
[CrossRef]

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954).
[CrossRef] [PubMed]

1948 (1)

1942 (1)

S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
[CrossRef] [PubMed]

1920 (1)

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” J. Royal Stat. Soc. A 83255–279 (1920).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[CrossRef]

Armstrong, J. A.

A. W. Smith, J. A. Armstrong, “Observation of photon counting distribution for laser light below threshold,” Phys. Lett. 19, 650–651 (1966).
[CrossRef]

Barlow, H. B.

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. 136, 469–488 (1957).
[PubMed]

H. B. Barlow, “Retinal noise and absolute threshold,” J. Opt. Soc. Am. 46, 634–639 (1956).
[CrossRef] [PubMed]

Birdsall, T. G.

W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
[CrossRef]

Egan, J. P.

J. P. Egan, Signal Detection Theory and ROC Analysis (Academic, New York, 1975).

Flint, G. W.

G. W. Flint, “Analysis and optimization of laser ranging techniques,” IEEE Trans. Mil. Electron. MIL-8, 22–28 (1964).
[CrossRef]

Fox, W. C.

W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
[CrossRef]

Freed, C.

C. Freed, H. A. Haus, “Photoelectron statistics produced by a laser operating below the threshold of oscillation,” Phys. Rev. Lett. 15, 943–946 (1965);“Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. QE-2, 190–195 (1966).
[CrossRef]

Gagliardi, R. M.

This calculation may also be carried out by using a special case of Eq. (7. 6. 2) along with the identity (C. 3. 1), in R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Goodman, J. W.

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, New York, 1965), 4th ed., p. xxxii.

Greenwood, M.

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” J. Royal Stat. Soc. A 83255–279 (1920).
[CrossRef]

Haus, H. A.

C. Freed, H. A. Haus, “Photoelectron statistics produced by a laser operating below the threshold of oscillation,” Phys. Rev. Lett. 15, 943–946 (1965);“Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. QE-2, 190–195 (1966).
[CrossRef]

Hecht, S.

S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
[CrossRef] [PubMed]

S. Hecht, “Energy and vision,” in Science in Progress, G. A. Baitsell, ed. (Yale U. P., New Haven, 1945), fourth series, pp. 75–95, 309–310.

Karp, S.

This calculation may also be carried out by using a special case of Eq. (7. 6. 2) along with the identity (C. 3. 1), in R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of photo-electrons,” Proc. Phys. Soc. 74, 233–242 (1959).
[CrossRef]

Martienssen, W.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);“Intensity fluctuations in light beams with several degrees of freedom,” Phys. Rev. Lett. 16, 531–533 (1966).
[CrossRef]

McGill, W. J.

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).Phys. Rev. Lett. 36, 1473 (1976).
[CrossRef]

W. J. McGill, “Neural counting mechanisms and energy detection in audition,” J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

Peterson, W. W.

W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
[CrossRef]

Pirenne, M. H.

S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
[CrossRef] [PubMed]

Prucnal, P. R.

M. C. Teich, P. R. Prucnal, “Negative binomial vs Poisson photon counting in visual psychophysics” (to be published).

M. C. Teich, P. R. Prucnal, “Photon counting and energy detection in vision” (to be published).

Reiffen, B.

B. Reiffen, H. Sherman, “An optimum demodulator for Poisson processes: photon source detectors,” Proc. IEEE 51, 1316–1320 (1963).
[CrossRef]

Rose, A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, New York, 1965), 4th ed., p. xxxii.

Sakitt, B.

B. Sakitt, “Canonical ratings,” Percept. Psychophys. 16, 478–488 (1974).
[CrossRef]

B. Sakitt, “Counting every quantum,” J. Physiol. 223, 131–150 (1972).
[PubMed]

Sherman, H.

B. Reiffen, H. Sherman, “An optimum demodulator for Poisson processes: photon source detectors,” Proc. IEEE 51, 1316–1320 (1963).
[CrossRef]

Shlaer, S.

S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
[CrossRef] [PubMed]

Smith, A. W.

A. W. Smith, J. A. Armstrong, “Observation of photon counting distribution for laser light below threshold,” Phys. Lett. 19, 650–651 (1966).
[CrossRef]

Spiller, E.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);“Intensity fluctuations in light beams with several degrees of freedom,” Phys. Rev. Lett. 16, 531–533 (1966).
[CrossRef]

Swets, J. A.

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954).
[CrossRef] [PubMed]

W. P. Tanner, J. A. Swets, “The human use of information—I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory IT-4, 213–221 (1954).
[CrossRef]

Tanner, W. P.

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. NY Acad. Sci. 89, 752–765 (1961).
[CrossRef]

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954).
[CrossRef] [PubMed]

W. P. Tanner, J. A. Swets, “The human use of information—I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory IT-4, 213–221 (1954).
[CrossRef]

Teich, M. C.

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).Phys. Rev. Lett. 36, 1473 (1976).
[CrossRef]

M. C. Teich, R. Y. Yen, “Photocounting receiver performance for detection of multimode laser or scattered radiation,” IEEE Trans. Aerosp. Electron. Syst. AES-8, 13–18 (1972).
[CrossRef]

M. C. Teich, P. R. Prucnal, “Photon counting and energy detection in vision” (to be published).

M. C. Teich, P. R. Prucnal, “Negative binomial vs Poisson photon counting in visual psychophysics” (to be published).

Van Trees, H. L.

H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968), Part I.

Yen, R. Y.

M. C. Teich, R. Y. Yen, “Photocounting receiver performance for detection of multimode laser or scattered radiation,” IEEE Trans. Aerosp. Electron. Syst. AES-8, 13–18 (1972).
[CrossRef]

Yule, G. U.

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” J. Royal Stat. Soc. A 83255–279 (1920).
[CrossRef]

Am. J. Phys. (1)

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);“Intensity fluctuations in light beams with several degrees of freedom,” Phys. Rev. Lett. 16, 531–533 (1966).
[CrossRef]

Ann. NY Acad. Sci. (1)

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. NY Acad. Sci. 89, 752–765 (1961).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

M. C. Teich, R. Y. Yen, “Photocounting receiver performance for detection of multimode laser or scattered radiation,” IEEE Trans. Aerosp. Electron. Syst. AES-8, 13–18 (1972).
[CrossRef]

IEEE Trans. Mil. Electron. (1)

G. W. Flint, “Analysis and optimization of laser ranging techniques,” IEEE Trans. Mil. Electron. MIL-8, 22–28 (1964).
[CrossRef]

IRE Trans. Inf. Theory (2)

W. W. Peterson, T. G. Birdsall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theory IT-4, 171–212 (1954).
[CrossRef]

W. P. Tanner, J. A. Swets, “The human use of information—I. Signal detection for the case of the signal known exactly,” IRE Trans. Inf. Theory IT-4, 213–221 (1954).
[CrossRef]

J. Physiol. (1)

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. 136, 469–488 (1957).
[PubMed]

J. Gen. Physiol. (1)

S. Hecht, S. Shlaer, M. H. Pirenne, “Energy, quanta and vision,” J. Gen. Physiol. 25, 819–840 (1942).
[CrossRef] [PubMed]

J. Math. Psychol. (1)

W. J. McGill, “Neural counting mechanisms and energy detection in audition,” J. Math. Psychol. 4, 351–376 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Physiol. (1)

B. Sakitt, “Counting every quantum,” J. Physiol. 223, 131–150 (1972).
[PubMed]

J. Royal Stat. Soc. A (1)

M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” J. Royal Stat. Soc. A 83255–279 (1920).
[CrossRef]

Percept. Psychophys. (1)

B. Sakitt, “Canonical ratings,” Percept. Psychophys. 16, 478–488 (1974).
[CrossRef]

Phys. Lett. (1)

A. W. Smith, J. A. Armstrong, “Observation of photon counting distribution for laser light below threshold,” Phys. Lett. 19, 650–651 (1966).
[CrossRef]

Phys. Rev. Lett. (1)

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[CrossRef]

Phys. Rev. Lett. (2)

C. Freed, H. A. Haus, “Photoelectron statistics produced by a laser operating below the threshold of oscillation,” Phys. Rev. Lett. 15, 943–946 (1965);“Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. QE-2, 190–195 (1966).
[CrossRef]

M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976).Phys. Rev. Lett. 36, 1473 (1976).
[CrossRef]

Proc. IEEE (1)

B. Reiffen, H. Sherman, “An optimum demodulator for Poisson processes: photon source detectors,” Proc. IEEE 51, 1316–1320 (1963).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Proc. Phys. Soc. (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photo-electrons,” Proc. Phys. Soc. 74, 233–242 (1959).
[CrossRef]

Psychol. Rev. (1)

W. P. Tanner, J. A. Swets, “A decision-making theory of visual detection,” Psychol. Rev. 61, 401–409 (1954).
[CrossRef] [PubMed]

Other (7)

S. Hecht, “Energy and vision,” in Science in Progress, G. A. Baitsell, ed. (Yale U. P., New Haven, 1945), fourth series, pp. 75–95, 309–310.

M. C. Teich, P. R. Prucnal, “Photon counting and energy detection in vision” (to be published).

M. C. Teich, P. R. Prucnal, “Negative binomial vs Poisson photon counting in visual psychophysics” (to be published).

H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968), Part I.

J. P. Egan, Signal Detection Theory and ROC Analysis (Academic, New York, 1975).

This calculation may also be carried out by using a special case of Eq. (7. 6. 2) along with the identity (C. 3. 1), in R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Yu. V. Geronimus, M. Yu. Tseytlin, eds. (Academic, New York, 1965), 4th ed., p. xxxii.

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Equations (15)

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Λ ( n ) = p ( n | H 1 ) p ( n | H 0 ) H 1 < H 0 λ ,
Λ ( n ) = [ k = 0 n p S ( k ) p D ( n k ) ] / p D ( n ) .
d 2 [ log f D ( x ) ] / d x 2 0 ,
d [ log f D ( x ) ] / d x d [ log f D ( x k ) ] / d x ,
f D ( x ) [ d f D ( x k ) / d x ] f D ( x k ) [ d f D ( x ) / d x ] 0 .
d [ p S ( k ) f D ( x k ) / f D ( x ) ] / d x 0 ,
Λ ( n + 1 ) Λ ( n ) = p S ( n + 1 ) p D ( 0 ) / p D ( n + 1 ) + k = 0 n p S ( k ) p D ( n + 1 k ) / p D ( n + 1 ) k = 0 n p S ( k ) p D ( n k ) / p D ( n ) ,
Λ ( n + 1 ) Λ ( n ) = p S ( n + 1 ) p D ( 0 ) / p D ( n + 1 ) + k = 0 n [ p S ( k ) f D ( n + 1 k ) / f D ( n + 1 ) p S ( k ) f D ( n k ) / f D ( n ) ] .
Λ ( n + 1 ) Λ ( n ) 0 ,
n H 1 < H 0 n t
p S ( n ) = Γ ( n + M ) n ! Γ ( M ) ( 1 + M n 0 ) n × ( 1 + n 0 M ) M .
p D ( n ) = p ( n | H 0 ) = n D n exp ( n D ) / n ! ;
d 2 [ ln f D ( x ) ] / d x 2 = n = 1 ( x + n ) 2 < 0 ,
p F = n = n t p ( n | H 0 ) = exp ( n D ) n = n t n D n / n ! α .
P D = n = n t p ( n | H 1 ) = n = n t k = 0 n [ n D n k exp ( n D ) ( n k ) ! × Γ ( k + M ) k ! Γ ( M ) ( 1 + M n 0 ) k ( 1 + n 0 M ) M ] .

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